Answer
No, the infinite geometric series does not have a limit.
Work Step by Step
The provided series is $2+3+\frac{9}{2}+\cdots $.
Here, ${{a}_{1}}=2$, $n=\infty $ and $r=\frac{3}{2}$
Since $\left| r \right|$ is not less than 1, ${{S}_{\infty }}$doesn’t exist.
Thus, the infinite geometric series $3+15+75+\cdots $ does not have a limit.
